I am working towards quantifying something I've said several times (e.g. here or here or here) -- that the trend towards lower growth as economies grow is a result of there being more ways an economy can be composed of many low growth industries than a few high growth industries, the former being a higher entropy state. This was an analogy with thermodynamics: there are more ways to emit several low energy photons than a few high energy ones, hence hot things are red, not blue (unless you get really hot ... )

Essentially, I am trying to derive the "economic temperature"

*~ log M*in a more rigorous way. See the section of the paper on "statistical economics" for a starting point.
Let's say nominal ouptut (NGDP) is given by

*N(t) = N0 exp ρ t*

with growth rate

*ρ*. Let's say that output consists of several industries (or firms), each with their own growth rate*rᵢ*so that*N(t) = A₁ exp r₁ t + A₂ exp r₂ t + ...*

If the growth rates are small and we keep ourselves focused on the short run, then we can say

*ρ t*<< 1 and*rᵢ t*<< 1 so that*N0 ~ A₁ + A₂ + ...*

*ρ ~ (1/N0) (A₁ r₁ + A₂ r₂ + ... )*

Let's say

*N0*is an integer, but very large. Then the*Aᵢ*represent a partition of*N0*. For example, here's a partition of*N0*= 1000:
These partitions have a smooth distribution for

*N0*>> 1 (the red line, the Vershik-Kerov-Logan-Shepp limit shape). Note, I mentioned Ferrers diagrams at the very beginning of this blog. Here are several partitions together:
The red curve represents the most likely paritition of an economy of nominal output

*N0*into various industries -- so we have the distribution of*Aᵢ*. The overall growth rate is then proportional to the dot product*Σ Aᵢ rᵢ*... note that*Aᵢ rᵢ*is itself another partition, this time of*N0 ρ*.
So if

*Aᵢ*has the VKLS distribution and*Aᵢ rᵢ*also has the VKLS distribution, we should be able to work out the distribution of rᵢ (under the assumption that*rᵢ*and*Aᵢ*are independent, Gibrat's law) ......

**Update 11/5/2015:**

The last statements aren't really true because of the possibility for negative growth rates

*rᵢ*. The terms*Aᵢ rᵢ*are not a partition of*N0 ρ.*
From the title I thought that you might be talking about class. ;)

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